LMFDB - Embedded newform 342.9.d.a.37.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [342,9,Mod(37,342)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(342, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("342.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 342 = 2 \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	342.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$139.323484641$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.6
Root			$-23.4825i$ of defining polynomial
Character	$\chi$	$=$	342.37
Dual form			342.9.d.a.37.12

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -128.000 q^{4} +919.278 q^{5} -343.629 q^{7} +1448.15i q^{8} +O(q^{10})$	\(q-11.3137i q^{2} -128.000 q^{4} +919.278 q^{5} -343.629 q^{7} +1448.15i q^{8} -10400.4i q^{10} -4190.29 q^{11} -16507.2i q^{13} +3887.72i q^{14} +16384.0 q^{16} -5020.47 q^{17} +(39747.8 - 124112. i) q^{19} -117668. q^{20} +47407.7i q^{22} -350263. q^{23} +454447. q^{25} -186758. q^{26} +43984.5 q^{28} +484397. i q^{29} +263837. i q^{31} -185364. i q^{32} +56800.1i q^{34} -315891. q^{35} -2.12516e6i q^{37} +(-1.40416e6 - 449695. i) q^{38} +1.33126e6i q^{40} +3.03570e6i q^{41} -2.35738e6 q^{43} +536357. q^{44} +3.96277e6i q^{46} -727875. q^{47} -5.64672e6 q^{49} -5.14148e6i q^{50} +2.11293e6i q^{52} +6.39013e6i q^{53} -3.85204e6 q^{55} -497628. i q^{56} +5.48032e6 q^{58} -1.53982e7i q^{59} +963914. q^{61} +2.98497e6 q^{62} -2.09715e6 q^{64} -1.51747e7i q^{65} -3.33818e7i q^{67} +642620. q^{68} +3.57390e6i q^{70} +2.44350e7i q^{71} +2.36687e7 q^{73} -2.40434e7 q^{74} +(-5.08772e6 + 1.58863e7i) q^{76} +1.43991e6 q^{77} -5.45025e7i q^{79} +1.50615e7 q^{80} +3.43450e7 q^{82} -4.12252e7 q^{83} -4.61521e6 q^{85} +2.66707e7i q^{86} -6.06818e6i q^{88} -7.15445e7i q^{89} +5.67237e6i q^{91} +4.48337e7 q^{92} +8.23496e6i q^{94} +(3.65393e7 - 1.14093e8i) q^{95} +1.33289e8i q^{97} +6.38853e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	$ 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) $ 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/342\mathbb{Z}\right)^\times$.

$n$	$191$	$325$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	11.3137i		−	0.707107i
$2$		−	11.3137i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−128.000			−0.500000
$5$	919.278			1.47085			0.735423	−	0.677609i	$-0.236984\pi$
$5$	919.278			1.47085			0.735423	+	0.677609i	$0.236984\pi$
$6$		0			0
$7$	−343.629			−0.143119			−0.0715596	−	0.997436i	$-0.522798\pi$
$7$	−343.629			−0.143119			−0.0715596	+	0.997436i	$0.522798\pi$
$8$			1448.15i			0.353553i
$9$		0			0
$10$		−	10400.4i		−	1.04004i
$11$	−4190.29			−0.286202			−0.143101	−	0.989708i	$-0.545707\pi$
$11$	−4190.29			−0.286202			−0.143101	+	0.989708i	$0.545707\pi$
$12$		0			0
$13$		−	16507.2i		−	0.577964i	−0.957335	−	0.288982i	$-0.906683\pi$
$13$		−	16507.2i		−	0.577964i	0.957335	−	0.288982i	$-0.0933168\pi$
$14$			3887.72i			0.101201i
$15$		0			0
$16$	16384.0			0.250000
$17$	−5020.47			−0.0601103			−0.0300551	−	0.999548i	$-0.509568\pi$
$17$	−5020.47			−0.0601103			−0.0300551	+	0.999548i	$0.509568\pi$
$18$		0			0
$19$	39747.8	−	124112.i	0.304999	−	0.952353i
$19$	39747.8	−	124112.i	0.304999	−	0.952353i
$20$	−117668.			−0.735423
$21$		0			0
$22$			47407.7i			0.202376i
$23$	−350263.			−1.25165			−0.625825	−	0.779963i	$-0.715238\pi$
$23$	−350263.			−1.25165			−0.625825	+	0.779963i	$0.715238\pi$
$24$		0			0
$25$	454447.			1.16339
$26$	−186758.			−0.408682
$27$		0			0
$28$	43984.5			0.0715596
$29$			484397.i			0.684872i	0.939541	+	0.342436i	$0.111252\pi$
$29$			484397.i			0.684872i	−0.939541	+	0.342436i	$0.888748\pi$
$30$		0			0
$31$			263837.i			0.285686i	0.989745	+	0.142843i	$0.0456244\pi$
$31$			263837.i			0.285686i	−0.989745	+	0.142843i	$0.954376\pi$
$32$		−	185364.i		−	0.176777i
$33$		0			0
$34$			56800.1i			0.0425044i
$35$	−315891.			−0.210506
$36$		0			0
$37$		−	2.12516e6i		−	1.13393i	−0.823743	−	0.566963i	$-0.808118\pi$
$37$		−	2.12516e6i		−	1.13393i	0.823743	−	0.566963i	$-0.191882\pi$
$38$	−1.40416e6	−	449695.i	−0.673415	−	0.215667i
$39$		0			0
$40$			1.33126e6i			0.520022i
$41$			3.03570e6i			1.07429i	0.843488	+	0.537147i	$0.180498\pi$
$41$			3.03570e6i			1.07429i	−0.843488	+	0.537147i	$0.819502\pi$
$42$		0			0
$43$	−2.35738e6			−0.689533			−0.344767	−	0.938688i	$-0.612042\pi$
$43$	−2.35738e6			−0.689533			−0.344767	+	0.938688i	$0.612042\pi$
$44$	536357.			0.143101
$45$		0			0
$46$			3.96277e6i			0.885051i
$47$	−727875.			−0.149164			−0.0745822	−	0.997215i	$-0.523762\pi$
$47$	−727875.			−0.149164			−0.0745822	+	0.997215i	$0.523762\pi$
$48$		0			0
$49$	−5.64672e6			−0.979517
$50$		−	5.14148e6i		−	0.822638i
$51$		0			0
$52$			2.11293e6i			0.288982i
$53$			6.39013e6i			0.809853i	0.914349	+	0.404927i	$0.132703\pi$
$53$			6.39013e6i			0.809853i	−0.914349	+	0.404927i	$0.867297\pi$
$54$		0			0
$55$	−3.85204e6			−0.420959
$56$		−	497628.i		−	0.0506003i
$57$		0			0
$58$	5.48032e6			0.484277
$59$		−	1.53982e7i		−	1.27075i	−0.772203	−	0.635376i	$-0.780845\pi$
$59$		−	1.53982e7i		−	1.27075i	0.772203	−	0.635376i	$-0.219155\pi$
$60$		0			0
$61$	963914.			0.0696176			0.0348088	−	0.999394i	$-0.488918\pi$
$61$	963914.			0.0696176			0.0348088	+	0.999394i	$0.488918\pi$
$62$	2.98497e6			0.202010
$63$		0			0
$64$	−2.09715e6			−0.125000
$65$		−	1.51747e7i		−	0.850095i
$66$		0			0
$67$		−	3.33818e7i		−	1.65657i	−0.560306	−	0.828286i	$-0.689316\pi$
$67$		−	3.33818e7i		−	1.65657i	0.560306	−	0.828286i	$-0.310684\pi$
$68$	642620.			0.0300551
$69$		0			0
$70$			3.57390e6i			0.148850i
$71$			2.44350e7i			0.961567i	0.876839	+	0.480783i	$0.159648\pi$
$71$			2.44350e7i			0.961567i	−0.876839	+	0.480783i	$0.840352\pi$
$72$		0			0
$73$	2.36687e7			0.833455			0.416728	−	0.909031i	$-0.363177\pi$
$73$	2.36687e7			0.833455			0.416728	+	0.909031i	$0.363177\pi$
$74$	−2.40434e7			−0.801807
$75$		0			0
$76$	−5.08772e6	+	1.58863e7i	−0.152500	+	0.476176i
$77$	1.43991e6			0.0409610
$78$		0			0
$79$		−	5.45025e7i		−	1.39929i	−0.714490	−	0.699646i	$-0.753341\pi$
$79$		−	5.45025e7i		−	1.39929i	0.714490	−	0.699646i	$-0.246659\pi$
$80$	1.50615e7			0.367711
$81$		0			0
$82$	3.43450e7			0.759641
$83$	−4.12252e7			−0.868662			−0.434331	−	0.900753i	$-0.643015\pi$
$83$	−4.12252e7			−0.868662			−0.434331	+	0.900753i	$0.643015\pi$
$84$		0			0
$85$	−4.61521e6			−0.0884129
$86$			2.66707e7i			0.487574i
$87$		0			0
$88$		−	6.06818e6i		−	0.101188i
$89$		−	7.15445e7i		−	1.14029i	−0.821543	−	0.570146i	$-0.806887\pi$
$89$		−	7.15445e7i		−	1.14029i	0.821543	−	0.570146i	$-0.193113\pi$
$90$		0			0
$91$			5.67237e6i			0.0827178i
$92$	4.48337e7			0.625825
$93$		0			0
$94$			8.23496e6i			0.105475i
$95$	3.65393e7	−	1.14093e8i	0.448607	−	1.40076i
$96$		0			0
$97$			1.33289e8i			1.50559i	0.658255	+	0.752795i	$0.271295\pi$
$97$			1.33289e8i			1.50559i	−0.658255	+	0.752795i	$0.728705\pi$
$98$			6.38853e7i			0.692623i
$99$		0			0
$100$	−5.81693e7			−0.581693
$101$	2.20883e7			0.212264			0.106132	−	0.994352i	$-0.466153\pi$
$101$	2.20883e7			0.212264			0.106132	+	0.994352i	$0.466153\pi$
$102$		0			0
$103$			1.42603e7i			0.126701i	0.997991	+	0.0633503i	$0.0201786\pi$
$103$			1.42603e7i			0.126701i	−0.997991	+	0.0633503i	$0.979821\pi$
$104$	2.39050e7			0.204341
$105$		0			0
$106$	7.22961e7			0.572653
$107$			5.24572e6i			0.0400193i	0.999800	+	0.0200097i	$0.00636970\pi$
$107$			5.24572e6i			0.0400193i	−0.999800	+	0.0200097i	$0.993630\pi$
$108$		0			0
$109$		−	1.90488e8i		−	1.34947i	−0.738061	−	0.674734i	$-0.764258\pi$
$109$		−	1.90488e8i		−	1.34947i	0.738061	−	0.674734i	$-0.235742\pi$
$110$			4.35809e7i			0.297663i
$111$		0			0
$112$	−5.63002e6			−0.0357798
$113$		−	2.04711e8i		−	1.25553i	−0.778403	−	0.627765i	$-0.783970\pi$
$113$		−	2.04711e8i		−	1.25553i	0.778403	−	0.627765i	$-0.216030\pi$
$114$		0			0
$115$	−3.21989e8			−1.84098
$116$		−	6.20028e7i		−	0.342436i
$117$		0			0
$118$	−1.74210e8			−0.898557
$119$	1.72518e6			0.00860293
$120$		0			0
$121$	−1.96800e8			−0.918088
$122$		−	1.09054e7i		−	0.0492271i
$123$		0			0
$124$		−	3.37711e7i		−	0.142843i
$125$	5.86705e7			0.240314
$126$		0			0
$127$			1.24877e7i			0.0480027i	0.999712	+	0.0240014i	$0.00764061\pi$
$127$			1.24877e7i			0.0480027i	−0.999712	+	0.0240014i	$0.992359\pi$
$128$			2.37266e7i			0.0883883i
$129$		0			0
$130$	−1.71683e8			−0.601108
$131$	−1.65537e8			−0.562094			−0.281047	−	0.959694i	$-0.590682\pi$
$131$	−1.65537e8			−0.562094			−0.281047	+	0.959694i	$0.590682\pi$
$132$		0			0
$133$	−1.36585e7	+	4.26484e7i	−0.0436512	+	0.136300i
$134$	−3.77672e8			−1.17137
$135$		0			0
$136$		−	7.27042e6i		−	0.0212522i
$137$	−5.08631e8			−1.44384			−0.721922	−	0.691974i	$-0.756741\pi$
$137$	−5.08631e8			−1.44384			−0.721922	+	0.691974i	$0.756741\pi$
$138$		0			0
$139$	1.29060e8			0.345727			0.172864	−	0.984946i	$-0.444698\pi$
$139$	1.29060e8			0.345727			0.172864	+	0.984946i	$0.444698\pi$
$140$	4.04340e7			0.105253
$141$		0			0
$142$	2.76451e8			0.679930
$143$			6.91700e7i			0.165415i
$144$		0			0
$145$			4.45295e8i			1.00734i
$146$		−	2.67780e8i		−	0.589342i
$147$		0			0
$148$			2.72021e8i			0.566963i
$149$	−7.74775e8			−1.57192			−0.785961	−	0.618277i	$-0.787831\pi$
$149$	−7.74775e8			−1.57192			−0.785961	+	0.618277i	$0.787831\pi$
$150$		0			0
$151$		−	7.13683e8i		−	1.37277i	−0.727238	−	0.686385i	$-0.759197\pi$
$151$		−	7.13683e8i		−	1.37277i	0.727238	−	0.686385i	$-0.240803\pi$
$152$	1.79733e8	+	5.75610e7i	0.336707	+	0.107833i
$153$		0			0
$154$		−	1.62907e7i		−	0.0289638i
$155$			2.42539e8i			0.420200i
$156$		0			0
$157$	−7.63715e8			−1.25699			−0.628497	−	0.777812i	$-0.716329\pi$
$157$	−7.63715e8			−1.25699			−0.628497	+	0.777812i	$0.716329\pi$
$158$	−6.16626e8			−0.989449
$159$		0			0
$160$		−	1.70401e8i		−	0.260011i
$161$	1.20361e8			0.179135
$162$		0			0
$163$	−5.73887e8			−0.812972			−0.406486	−	0.913657i	$-0.633246\pi$
$163$	−5.73887e8			−0.812972			−0.406486	+	0.913657i	$0.633246\pi$
$164$		−	3.88570e8i		−	0.537147i
$165$		0			0
$166$			4.66410e8i			0.614237i
$167$			5.55377e8i			0.714040i	0.934097	+	0.357020i	$0.116207\pi$
$167$			5.55377e8i			0.714040i	−0.934097	+	0.357020i	$0.883793\pi$
$168$		0			0
$169$	5.43242e8			0.665958
$170$			5.22151e7i			0.0625173i
$171$		0			0
$172$	3.01744e8			0.344767
$173$			6.56811e8i			0.733256i	0.930368	+	0.366628i	$0.119488\pi$
$173$			6.56811e8i			0.733256i	−0.930368	+	0.366628i	$0.880512\pi$
$174$		0			0
$175$	−1.56161e8			−0.166503
$176$	−6.86537e7			−0.0715506
$177$		0			0
$178$	−8.09434e8			−0.806309
$179$			1.17573e9i			1.14523i	0.819823	+	0.572617i	$0.194072\pi$
$179$			1.17573e9i			1.14523i	−0.819823	+	0.572617i	$0.805928\pi$
$180$		0			0
$181$			6.73071e8i			0.627114i	0.949569	+	0.313557i	$0.101521\pi$
$181$			6.73071e8i			0.627114i	−0.949569	+	0.313557i	$0.898479\pi$
$182$	6.41755e7			0.0584903
$183$		0			0
$184$		−	5.07235e8i		−	0.442525i
$185$		−	1.95361e9i		−	1.66783i
$186$		0			0
$187$	2.10372e7			0.0172037
$188$	9.31680e7			0.0745822
$189$		0			0
$190$	−1.29082e9	−	4.13395e8i	−0.990489	−	0.317213i
$191$	2.49800e9			1.87698			0.938488	−	0.345311i	$-0.112227\pi$
$191$	2.49800e9			1.87698			0.938488	+	0.345311i	$0.112227\pi$
$192$		0			0
$193$			2.38380e9i			1.71807i	0.511921	+	0.859033i	$0.328934\pi$
$193$			2.38380e9i			1.71807i	−0.511921	+	0.859033i	$0.671066\pi$
$194$	1.50799e9			1.06461
$195$		0			0
$196$	7.22780e8			0.489758
$197$	4.08382e8			0.271145			0.135573	−	0.990767i	$-0.456713\pi$
$197$	4.08382e8			0.271145			0.135573	+	0.990767i	$0.456713\pi$
$198$		0			0
$199$	−1.99524e9			−1.27228			−0.636139	−	0.771574i	$-0.719470\pi$
$199$	−1.99524e9			−1.27228			−0.636139	+	0.771574i	$0.719470\pi$
$200$			6.58110e8i			0.411319i
$201$		0			0
$202$		−	2.49900e8i		−	0.150093i
$203$		−	1.66453e8i		−	0.0980183i
$204$		0			0
$205$			2.79065e9i			1.58012i
$206$	1.61337e8			0.0895909
$207$		0			0
$208$		−	2.70454e8i		−	0.144491i
$209$	−1.66555e8	+	5.20063e8i	−0.0872915	+	0.272565i
$210$		0			0
$211$		−	2.21085e9i		−	1.11540i	−0.830044	−	0.557698i	$-0.811685\pi$
$211$		−	2.21085e9i		−	1.11540i	0.830044	−	0.557698i	$-0.188315\pi$
$212$		−	8.17937e8i		−	0.404927i
$213$		0			0
$214$	5.93485e7			0.0282979
$215$	−2.16708e9			−1.01420
$216$		0			0
$217$		−	9.06621e7i		−	0.0408871i
$218$	−2.15513e9			−0.954218
$219$		0			0
$220$	4.93061e8			0.210480
$221$			8.28740e7i			0.0347416i
$222$		0			0
$223$		−	1.00945e9i		−	0.408193i	−0.978951	−	0.204096i	$-0.934574\pi$
$223$		−	1.00945e9i		−	0.408193i	0.978951	−	0.204096i	$-0.0654256\pi$
$224$			6.36964e7i			0.0253001i
$225$		0			0
$226$	−2.31604e9			−0.887794
$227$		−	1.07230e9i		−	0.403844i	−0.979402	−	0.201922i	$-0.935281\pi$
$227$		−	1.07230e9i		−	0.403844i	0.979402	−	0.201922i	$-0.0647187\pi$
$228$		0			0
$229$	−2.18741e9			−0.795406			−0.397703	−	0.917514i	$-0.630193\pi$
$229$	−2.18741e9			−0.795406			−0.397703	+	0.917514i	$0.630193\pi$
$230$			3.64289e9i			1.30177i
$231$		0			0
$232$	−7.01481e8			−0.242139
$233$	1.60064e9			0.543087			0.271544	−	0.962426i	$-0.412466\pi$
$233$	1.60064e9			0.543087			0.271544	+	0.962426i	$0.412466\pi$
$234$		0			0
$235$	−6.69119e8			−0.219398
$236$			1.97096e9i			0.635376i
$237$		0			0
$238$		−	1.95182e7i		−	0.00608319i
$239$	−2.42499e9			−0.743222			−0.371611	−	0.928388i	$-0.621195\pi$
$239$	−2.42499e9			−0.743222			−0.371611	+	0.928388i	$0.621195\pi$
$240$		0			0
$241$			1.99574e9i			0.591610i	0.955248	+	0.295805i	$0.0955879\pi$
$241$			1.99574e9i			0.591610i	−0.955248	+	0.295805i	$0.904412\pi$
$242$			2.22654e9i			0.649186i
$243$		0			0
$244$	−1.23381e8			−0.0348088
$245$	−5.19091e9			−1.44072
$246$		0			0
$247$	−2.04874e9	−	6.56126e8i	−0.550426	−	0.176279i
$248$	−3.82077e8			−0.101005
$249$		0			0
$250$		−	6.63781e8i		−	0.169928i
$251$	−2.54465e9			−0.641110			−0.320555	−	0.947230i	$-0.603869\pi$
$251$	−2.54465e9			−0.641110			−0.320555	+	0.947230i	$0.603869\pi$
$252$		0			0
$253$	1.46770e9			0.358225
$254$	1.41282e8			0.0339431
$255$		0			0
$256$	2.68435e8			0.0625000
$257$			5.40032e9i			1.23790i	0.785429	+	0.618952i	$0.212442\pi$
$257$			5.40032e9i			1.23790i	−0.785429	+	0.618952i	$0.787558\pi$
$258$		0			0
$259$			7.30267e8i			0.162287i
$260$			1.94237e9i			0.425048i
$261$		0			0
$262$			1.87283e9i			0.397460i
$263$	−4.20312e9			−0.878514			−0.439257	−	0.898361i	$-0.644758\pi$
$263$	−4.20312e9			−0.878514			−0.439257	+	0.898361i	$0.644758\pi$
$264$		0			0
$265$			5.87431e9i			1.19117i
$266$	4.82511e8	+	1.54528e8i	0.0963786	+	0.0308661i
$267$		0			0
$268$			4.27287e9i			0.828286i
$269$			1.65507e7i			0.00316087i	0.999999	+	0.00158044i	$0.000503069\pi$
$269$			1.65507e7i			0.00316087i	−0.999999	+	0.00158044i	$0.999497\pi$
$270$		0			0
$271$	−8.10231e9			−1.50221			−0.751107	−	0.660180i	$-0.770480\pi$
$271$	−8.10231e9			−1.50221			−0.751107	+	0.660180i	$0.770480\pi$
$272$	−8.22554e7			−0.0150276
$273$		0			0
$274$			5.75450e9i			1.02095i
$275$	−1.90427e9			−0.332964
$276$		0			0
$277$	−4.75322e9			−0.807363			−0.403682	−	0.914900i	$-0.632270\pi$
$277$	−4.75322e9			−0.807363			−0.403682	+	0.914900i	$0.632270\pi$
$278$		−	1.46015e9i		−	0.244466i
$279$		0			0
$280$		−	4.57459e8i		−	0.0744252i
$281$			1.26345e9i			0.202643i	0.994854	+	0.101321i	$0.0323070\pi$
$281$			1.26345e9i			0.202643i	−0.994854	+	0.101321i	$0.967693\pi$
$282$		0			0
$283$	−6.11988e9			−0.954107			−0.477054	−	0.878874i	$-0.658295\pi$
$283$	−6.11988e9			−0.954107			−0.477054	+	0.878874i	$0.658295\pi$
$284$		−	3.12768e9i		−	0.480783i
$285$		0			0
$286$	7.82570e8			0.116966
$287$		−	1.04316e9i		−	0.153752i
$288$		0			0
$289$	−6.95055e9			−0.996387
$290$	5.03794e9			0.712297
$291$		0			0
$292$	−3.02959e9			−0.416728
$293$		−	8.14824e9i		−	1.10559i	−0.833318	−	0.552794i	$-0.813562\pi$
$293$		−	8.14824e9i		−	1.10559i	0.833318	−	0.552794i	$-0.186438\pi$
$294$		0			0
$295$		−	1.41552e10i		−	1.86908i
$296$	3.07756e9			0.400903
$297$		0			0
$298$			8.76558e9i			1.11152i
$299$			5.78187e9i			0.723409i
$300$		0			0
$301$	8.10063e8			0.0986854
$302$	−8.07440e9			−0.970695
$303$		0			0
$304$	6.51228e8	−	2.03344e9i	0.0762498	−	0.238088i
$305$	8.86105e8			0.102397
$306$		0			0
$307$		−	7.97886e9i		−	0.898229i	−0.893474	−	0.449115i	$-0.851739\pi$
$307$		−	7.97886e9i		−	0.898229i	0.893474	−	0.449115i	$-0.148261\pi$
$308$	−1.84308e8			−0.0204805
$309$		0			0
$310$	2.74402e9			0.297126
$311$	1.02790e10			1.09878			0.549390	−	0.835566i	$-0.314860\pi$
$311$	1.02790e10			1.09878			0.549390	+	0.835566i	$0.314860\pi$
$312$		0			0
$313$	6.84361e9			0.713030			0.356515	−	0.934290i	$-0.383965\pi$
$313$	6.84361e9			0.713030			0.356515	+	0.934290i	$0.383965\pi$
$314$			8.64045e9i			0.888828i
$315$		0			0
$316$			6.97632e9i			0.699646i
$317$		−	1.07817e10i		−	1.06770i	−0.845579	−	0.533851i	$-0.820744\pi$
$317$		−	1.07817e10i		−	1.06770i	0.845579	−	0.533851i	$-0.179256\pi$
$318$		0			0
$319$		−	2.02976e9i		−	0.196012i
$320$	−1.92787e9			−0.183856
$321$		0			0
$322$		−	1.36173e9i		−	0.126668i
$323$	−1.99553e8	+	6.23098e8i	−0.0183336	+	0.0572462i
$324$		0			0
$325$		−	7.50167e9i		−	0.672395i
$326$			6.49279e9i			0.574858i
$327$		0			0
$328$	−4.39616e9			−0.379821
$329$	2.50119e8			0.0213483
$330$		0			0
$331$		−	4.83111e9i		−	0.402471i	−0.979543	−	0.201235i	$-0.935504\pi$
$331$		−	4.83111e9i		−	0.402471i	0.979543	−	0.201235i	$-0.0644956\pi$
$332$	5.27683e9			0.434331
$333$		0			0
$334$	6.28338e9			0.504902
$335$		−	3.06871e10i		−	2.43656i
$336$		0			0
$337$		−	1.24834e10i		−	0.967860i	−0.875107	−	0.483930i	$-0.839209\pi$
$337$		−	1.24834e10i		−	0.967860i	0.875107	−	0.483930i	$-0.160791\pi$
$338$		−	6.14608e9i		−	0.470903i
$339$		0			0
$340$	5.90747e8			0.0442064
$341$		−	1.10555e9i		−	0.0817639i
$342$		0			0
$343$	3.92133e9			0.283307
$344$		−	3.41385e9i		−	0.243787i
$345$		0			0
$346$	7.43096e9			0.518490
$347$	−2.24032e10			−1.54522			−0.772612	−	0.634879i	$-0.781050\pi$
$347$	−2.24032e10			−1.54522			−0.772612	+	0.634879i	$0.781050\pi$
$348$		0			0
$349$	2.47555e10			1.66867			0.834335	−	0.551257i	$-0.185852\pi$
$349$	2.47555e10			1.66867			0.834335	+	0.551257i	$0.185852\pi$
$350$			1.76676e9i			0.117735i
$351$		0			0
$352$			7.76728e8i			0.0505939i
$353$	−7.58628e9			−0.488573			−0.244287	−	0.969703i	$-0.578554\pi$
$353$	−7.58628e9			−0.488573			−0.244287	+	0.969703i	$0.578554\pi$
$354$		0			0
$355$			2.24626e10i			1.41432i
$356$			9.15770e9i			0.570146i
$357$		0			0
$358$	1.33018e10			0.809802
$359$	6.91505e9			0.416311			0.208155	−	0.978096i	$-0.433254\pi$
$359$	6.91505e9			0.416311			0.208155	+	0.978096i	$0.433254\pi$
$360$		0			0
$361$	−1.38238e10	−	9.86632e9i	−0.813951	−	0.580934i
$362$	7.61493e9			0.443436
$363$		0			0
$364$		−	7.26063e8i		−	0.0413589i
$365$	2.17581e10			1.22588
$366$		0			0
$367$	3.46443e10			1.90971			0.954855	−	0.297072i	$-0.0960101\pi$
$367$	3.46443e10			1.90971			0.954855	+	0.297072i	$0.0960101\pi$
$368$	−5.73871e9			−0.312913
$369$		0			0
$370$	−2.21026e10			−1.17933
$371$		−	2.19584e9i		−	0.115906i
$372$		0			0
$373$			1.32696e10i			0.685521i	0.939423	+	0.342761i	$0.111362\pi$
$373$			1.32696e10i			0.685521i	−0.939423	+	0.342761i	$0.888638\pi$
$374$		−	2.38009e8i		−	0.0121648i
$375$		0			0
$376$		−	1.05408e9i		−	0.0527376i
$377$	7.99605e9			0.395831
$378$		0			0
$379$			1.37892e10i			0.668318i	0.942517	+	0.334159i	$0.108452\pi$
$379$			1.37892e10i			0.668318i	−0.942517	+	0.334159i	$0.891548\pi$
$380$	−4.67703e9	+	1.46039e10i	−0.224303	+	0.700382i
$381$		0			0
$382$		−	2.82616e10i		−	1.32722i
$383$		−	2.21901e10i		−	1.03125i	−0.856814	−	0.515626i	$-0.827560\pi$
$383$		−	2.21901e10i		−	1.03125i	0.856814	−	0.515626i	$-0.172440\pi$
$384$		0			0
$385$	1.32367e9			0.0602474
$386$	2.69696e10			1.21486
$387$		0			0
$388$		−	1.70610e10i		−	0.752795i
$389$	−5.26015e9			−0.229721			−0.114860	−	0.993382i	$-0.536642\pi$
$389$	−5.26015e9			−0.229721			−0.114860	+	0.993382i	$0.536642\pi$
$390$		0			0
$391$	1.75848e9			0.0752370
$392$		−	8.17732e9i		−	0.346312i
$393$		0			0
$394$		−	4.62032e9i		−	0.191729i
$395$		−	5.01030e10i		−	2.05814i
$396$		0			0
$397$	−8.36911e9			−0.336913			−0.168456	−	0.985709i	$-0.553878\pi$
$397$	−8.36911e9			−0.336913			−0.168456	+	0.985709i	$0.553878\pi$
$398$			2.25735e10i			0.899637i
$399$		0			0
$400$	7.44567e9			0.290846
$401$			3.79985e10i			1.46957i	0.678302	+	0.734783i	$0.262716\pi$
$401$			3.79985e10i			1.46957i	−0.678302	+	0.734783i	$0.737284\pi$
$402$		0			0
$403$	4.35522e9			0.165116
$404$	−2.82730e9			−0.106132
$405$		0			0
$406$	−1.88320e9			−0.0693094
$407$			8.90503e9i			0.324532i
$408$		0			0
$409$		−	4.18057e9i		−	0.149397i	−0.997206	−	0.0746985i	$-0.976201\pi$
$409$		−	4.18057e9i		−	0.149397i	0.997206	−	0.0746985i	$-0.0237994\pi$
$410$	3.15726e10			1.11731
$411$		0			0
$412$		−	1.82532e9i		−	0.0633503i
$413$			5.29126e9i			0.181869i
$414$		0			0
$415$	−3.78975e10			−1.27767
$416$	−3.05984e9			−0.102171
$417$		0			0
$418$	5.88384e9	+	1.88435e9i	0.192733	+	0.0617244i
$419$	−6.81207e9			−0.221016			−0.110508	−	0.993875i	$-0.535248\pi$
$419$	−6.81207e9			−0.221016			−0.110508	+	0.993875i	$0.535248\pi$
$420$		0			0
$421$			4.75587e9i			0.151392i	0.997131	+	0.0756958i	$0.0241178\pi$
$421$			4.75587e9i			0.151392i	−0.997131	+	0.0756958i	$0.975882\pi$
$422$	−2.50129e10			−0.788704
$423$		0			0
$424$	−9.25390e9			−0.286326
$425$	−2.28154e9			−0.0699314
$426$		0			0
$427$	−3.31229e8			−0.00996362
$428$		−	6.71452e8i		−	0.0200097i
$429$		0			0
$430$			2.45178e10i			0.717145i
$431$		−	5.49100e10i		−	1.59126i	−0.605780	−	0.795632i	$-0.707139\pi$
$431$		−	5.49100e10i		−	1.59126i	0.605780	−	0.795632i	$-0.292861\pi$
$432$		0			0
$433$			5.67002e10i			1.61300i	0.591237	+	0.806498i	$0.298640\pi$
$433$			5.67002e10i			1.61300i	−0.591237	+	0.806498i	$0.701360\pi$
$434$	−1.02572e9			−0.0289116
$435$		0			0
$436$			2.43825e10i			0.674734i
$437$	−1.39222e10	+	4.34717e10i	−0.381752	+	1.19201i
$438$		0			0
$439$		−	6.67486e10i		−	1.79715i	−0.438822	−	0.898574i	$-0.644604\pi$
$439$		−	6.67486e10i		−	1.79715i	0.438822	−	0.898574i	$-0.355396\pi$
$440$		−	5.57835e9i		−	0.148832i
$441$		0			0
$442$	9.37613e8			0.0245660
$443$	4.33866e9			0.112653			0.0563263	−	0.998412i	$-0.482061\pi$
$443$	4.33866e9			0.112653			0.0563263	+	0.998412i	$0.482061\pi$
$444$		0			0
$445$		−	6.57693e10i		−	1.67719i
$446$	−1.14206e10			−0.288636
$447$		0			0
$448$	7.20643e8			0.0178899
$449$		−	7.17821e10i		−	1.76616i	−0.469219	−	0.883082i	$-0.655464\pi$
$449$		−	7.17821e10i		−	1.76616i	0.469219	−	0.883082i	$-0.344536\pi$
$450$		0			0
$451$		−	1.27205e10i		−	0.307466i
$452$			2.62030e10i			0.627765i
$453$		0			0
$454$	−1.21317e10			−0.285561
$455$			5.21448e9i			0.121665i
$456$		0			0
$457$	8.06293e10			1.84854			0.924268	−	0.381743i	$-0.124676\pi$
$457$	8.06293e10			1.84854			0.924268	+	0.381743i	$0.124676\pi$
$458$			2.47478e10i			0.562437i
$459$		0			0
$460$	4.12146e10			0.920492
$461$	5.53201e9			0.122484			0.0612420	−	0.998123i	$-0.480494\pi$
$461$	5.53201e9			0.122484			0.0612420	+	0.998123i	$0.480494\pi$
$462$		0			0
$463$	4.32453e10			0.941054			0.470527	−	0.882386i	$-0.344064\pi$
$463$	4.32453e10			0.941054			0.470527	+	0.882386i	$0.344064\pi$
$464$			7.93636e9i			0.171218i
$465$		0			0
$466$		−	1.81092e10i		−	0.384021i
$467$	−2.49413e10			−0.524386			−0.262193	−	0.965015i	$-0.584446\pi$
$467$	−2.49413e10			−0.524386			−0.262193	+	0.965015i	$0.584446\pi$
$468$		0			0
$469$			1.14710e10i			0.237087i
$470$			7.57022e9i			0.155138i
$471$		0			0
$472$	2.22989e10			0.449279
$473$	9.87809e9			0.197346
$474$		0			0
$475$	1.80633e10	−	5.64022e10i	0.354832	−	1.10795i
$476$	−2.20823e8			−0.00430147
$477$		0			0
$478$			2.74357e10i			0.525538i
$479$	−5.76944e10			−1.09595			−0.547977	−	0.836494i	$-0.684602\pi$
$479$	−5.76944e10			−1.09595			−0.547977	+	0.836494i	$0.684602\pi$
$480$		0			0
$481$	−3.50805e10			−0.655369
$482$	2.25792e10			0.418332
$483$		0			0
$484$	2.51904e10			0.459044
$485$			1.22529e11i			2.21449i
$486$		0			0
$487$			3.94004e10i			0.700462i	0.936663	+	0.350231i	$0.113897\pi$
$487$			3.94004e10i			0.700462i	−0.936663	+	0.350231i	$0.886103\pi$
$488$			1.39590e9i			0.0246135i
$489$		0			0
$490$			5.87284e10i			1.01874i
$491$	1.08173e11			1.86120			0.930599	−	0.366041i	$-0.119287\pi$
$491$	1.08173e11			1.86120			0.930599	+	0.366041i	$0.119287\pi$
$492$		0			0
$493$		−	2.43190e9i		−	0.0411678i
$494$	−7.42322e9	+	2.31788e10i	−0.124648	+	0.389210i
$495$		0			0
$496$			4.32270e9i			0.0714215i
$497$		−	8.39659e9i		−	0.137619i
$498$		0			0
$499$	8.97074e10			1.44686			0.723430	−	0.690398i	$-0.242565\pi$
$499$	8.97074e10			1.44686			0.723430	+	0.690398i	$0.242565\pi$
$500$	−7.50982e9			−0.120157
$501$		0			0
$502$			2.87894e10i			0.453333i
$503$	7.42237e10			1.15950			0.579750	−	0.814795i	$-0.303150\pi$
$503$	7.42237e10			1.15950			0.579750	+	0.814795i	$0.303150\pi$
$504$		0			0
$505$	2.03053e10			0.312207
$506$		−	1.66052e10i		−	0.253303i
$507$		0			0
$508$		−	1.59842e9i		−	0.0240014i
$509$			3.30960e10i			0.493066i	0.969134	+	0.246533i	$0.0792913\pi$
$509$			3.30960e10i			0.493066i	−0.969134	+	0.246533i	$0.920709\pi$
$510$		0			0
$511$	−8.13324e9			−0.119283
$512$		−	3.03700e9i		−	0.0441942i
$513$		0			0
$514$	6.10976e10			0.875330
$515$			1.31092e10i			0.186357i
$516$		0			0
$517$	3.05000e9			0.0426912
$518$	8.26203e9			0.114754
$519$		0			0
$520$	2.19754e10			0.300554
$521$		−	1.15808e11i		−	1.57177i	−0.618372	−	0.785886i	$-0.712207\pi$
$521$		−	1.15808e11i		−	1.57177i	0.618372	−	0.785886i	$-0.287793\pi$
$522$		0			0
$523$		−	1.39644e11i		−	1.86644i	−0.359305	−	0.933220i	$-0.616986\pi$
$523$		−	1.39644e11i		−	1.86644i	0.359305	−	0.933220i	$-0.383014\pi$
$524$	2.11887e10			0.281047
$525$		0			0
$526$			4.75529e10i			0.621204i
$527$		−	1.32458e9i		−	0.0171726i
$528$		0			0
$529$	4.43733e10			0.566629
$530$	6.64602e10			0.842283
$531$		0			0
$532$	1.74829e9	−	5.45899e9i	0.0218256	−	0.0681500i
$533$	5.01110e10			0.620904
$534$		0			0
$535$			4.82227e9i			0.0588622i
$536$	4.83420e10			0.585687
$537$		0			0
$538$	1.87250e8			0.00223508
$539$	2.36614e10			0.280340
$540$		0			0
$541$	−1.09728e11			−1.28094			−0.640470	−	0.767983i	$-0.721260\pi$
$541$	−1.09728e11			−1.28094			−0.640470	+	0.767983i	$0.721260\pi$
$542$			9.16672e10i			1.06223i
$543$		0			0
$544$			9.30613e8i			0.0106261i
$545$		−	1.75112e11i		−	1.98486i
$546$		0			0
$547$			7.70367e10i			0.860495i	0.902711	+	0.430247i	$0.141574\pi$
$547$			7.70367e10i			0.860495i	−0.902711	+	0.430247i	$0.858426\pi$
$548$	6.51047e10			0.721922
$549$		0			0
$550$			2.15443e10i			0.235441i
$551$	6.01192e10	+	1.92537e10i	0.652239	+	0.208885i
$552$		0			0
$553$			1.87287e10i			0.200266i
$554$			5.37766e10i			0.570892i
$555$		0			0
$556$	−1.65197e10			−0.172864
$557$	8.33803e10			0.866248			0.433124	−	0.901334i	$-0.357411\pi$
$557$	8.33803e10			0.866248			0.433124	+	0.901334i	$0.357411\pi$
$558$		0			0
$559$			3.89138e10i			0.398525i
$560$	−5.17556e9			−0.0526265
$561$		0			0
$562$	1.42943e10			0.143290
$563$		−	1.15774e11i		−	1.15233i	−0.817334	−	0.576164i	$-0.804549\pi$
$563$		−	1.15774e11i		−	1.15233i	0.817334	−	0.576164i	$-0.195451\pi$
$564$		0			0
$565$		−	1.88186e11i		−	1.84669i
$566$			6.92386e10i			0.674656i
$567$		0			0
$568$	−3.53857e10			−0.339965
$569$		−	7.99151e10i		−	0.762394i	−0.924494	−	0.381197i	$-0.875512\pi$
$569$		−	7.99151e10i		−	0.762394i	0.924494	−	0.381197i	$-0.124488\pi$
$570$		0			0
$571$	−1.62039e11			−1.52432			−0.762159	−	0.647390i	$-0.775861\pi$
$571$	−1.62039e11			−1.52432			−0.762159	+	0.647390i	$0.775861\pi$
$572$		−	8.85377e9i		−	0.0827073i
$573$		0			0
$574$	−1.18020e10			−0.108719
$575$	−1.59176e11			−1.45615
$576$		0			0
$577$	1.78544e11			1.61080			0.805402	−	0.592728i	$-0.201949\pi$
$577$	1.78544e11			1.61080			0.805402	+	0.592728i	$0.201949\pi$
$578$			7.86365e10i			0.704552i
$579$		0			0
$580$		−	5.69978e10i		−	0.503670i
$581$	1.41662e10			0.124322
$582$		0			0
$583$		−	2.67765e10i		−	0.231782i
$584$			3.42759e10i			0.294671i
$585$		0			0
$586$	−9.21868e10			−0.781768
$587$	−2.06207e11			−1.73681			−0.868404	−	0.495857i	$-0.834854\pi$
$587$	−2.06207e11			−1.73681			−0.868404	+	0.495857i	$0.834854\pi$
$588$		0			0
$589$	3.27452e10	+	1.04869e10i	0.272074	+	0.0871339i
$590$	−1.60148e11			−1.32164
$591$		0			0
$592$		−	3.48186e10i		−	0.283482i
$593$	8.30747e10			0.671815			0.335908	−	0.941895i	$-0.390957\pi$
$593$	8.30747e10			0.671815			0.335908	+	0.941895i	$0.390957\pi$
$594$		0			0
$595$	1.58592e9			0.0126536
$596$	9.91712e10			0.785961
$597$		0			0
$598$	6.54144e10			0.511527
$599$			2.05057e11i			1.59282i	0.604757	+	0.796410i	$0.293270\pi$
$599$			2.05057e11i			1.59282i	−0.604757	+	0.796410i	$0.706730\pi$
$600$		0			0
$601$			1.04583e11i			0.801611i	0.916163	+	0.400805i	$0.131270\pi$
$601$			1.04583e11i			0.801611i	−0.916163	+	0.400805i	$0.868730\pi$
$602$		−	9.16482e9i		−	0.0697811i
$603$		0			0
$604$			9.13515e10i			0.686385i
$605$	−1.80914e11			−1.35037
$606$		0			0
$607$			1.21178e11i			0.892626i	0.894877	+	0.446313i	$0.147263\pi$
$607$			1.21178e11i			0.892626i	−0.894877	+	0.446313i	$0.852737\pi$
$608$	−2.30058e10	−	7.36780e9i	−0.168354	−	0.0539167i
$609$		0			0
$610$		−	1.00251e10i		−	0.0724054i
$611$			1.20152e10i			0.0862117i
$612$		0			0
$613$	−5.78476e10			−0.409678			−0.204839	−	0.978796i	$-0.565667\pi$
$613$	−5.78476e10			−0.409678			−0.204839	+	0.978796i	$0.565667\pi$
$614$	−9.02705e10			−0.635144
$615$		0			0
$616$			2.08521e9i			0.0144819i
$617$	−1.37356e11			−0.947776			−0.473888	−	0.880585i	$-0.657150\pi$
$617$	−1.37356e11			−0.947776			−0.473888	+	0.880585i	$0.657150\pi$
$618$		0			0
$619$	−8.69099e10			−0.591979			−0.295990	−	0.955191i	$-0.595649\pi$
$619$	−8.69099e10			−0.591979			−0.295990	+	0.955191i	$0.595649\pi$
$620$		−	3.10451e10i		−	0.210100i
$621$		0			0
$622$		−	1.16294e11i		−	0.776955i
$623$			2.45848e10i			0.163198i
$624$		0			0
$625$	−1.23584e11			−0.809920
$626$		−	7.74266e10i		−	0.504189i
$627$		0			0
$628$	9.77556e10			0.628497
$629$			1.06693e10i			0.0681606i
$630$		0			0
$631$	1.40125e11			0.883889			0.441945	−	0.897042i	$-0.354289\pi$
$631$	1.40125e11			0.883889			0.441945	+	0.897042i	$0.354289\pi$
$632$	7.89281e10			0.494724
$633$		0			0
$634$	−1.21981e11			−0.754979
$635$			1.14796e10i			0.0706046i
$636$		0			0
$637$			9.32117e10i			0.566125i
$638$	−2.29641e10			−0.138601
$639$		0			0
$640$			2.18113e10i			0.130006i
$641$		−	2.18831e11i		−	1.29621i	−0.761549	−	0.648107i	$-0.775561\pi$
$641$		−	2.18831e11i		−	1.29621i	0.761549	−	0.648107i	$-0.224439\pi$
$642$		0			0
$643$	6.72532e10			0.393431			0.196716	−	0.980461i	$-0.436972\pi$
$643$	6.72532e10			0.393431			0.196716	+	0.980461i	$0.436972\pi$
$644$	−1.54062e10			−0.0895676
$645$		0			0
$646$	7.04955e9	+	2.25768e9i	0.0404791	+	0.0129638i
$647$	−2.70481e9			−0.0154355			−0.00771773	−	0.999970i	$-0.502457\pi$
$647$	−2.70481e9			−0.0154355			−0.00771773	+	0.999970i	$0.502457\pi$
$648$		0			0
$649$			6.45227e10i			0.363692i
$650$	−8.48717e10			−0.475455
$651$		0			0
$652$	7.34575e10			0.406486
$653$	−2.53210e11			−1.39260			−0.696302	−	0.717749i	$-0.745172\pi$
$653$	−2.53210e11			−1.39260			−0.696302	+	0.717749i	$0.745172\pi$
$654$		0			0
$655$	−1.52174e11			−0.826753
$656$			4.97369e10i			0.268574i
$657$		0			0
$658$		−	2.82977e9i		−	0.0150955i
$659$			1.31513e9i			0.00697310i	0.999994	+	0.00348655i	$0.00110981\pi$
$659$			1.31513e9i			0.00697310i	−0.999994	+	0.00348655i	$0.998890\pi$
$660$		0			0
$661$			1.05346e11i			0.551836i	0.961181	+	0.275918i	$0.0889819\pi$
$661$			1.05346e11i			0.551836i	−0.961181	+	0.275918i	$0.911018\pi$
$662$	−5.46577e10			−0.284590
$663$		0			0
$664$		−	5.97005e10i		−	0.307118i
$665$	−1.25560e10	+	3.92057e10i	−0.0642042	+	0.200476i
$666$		0			0
$667$		−	1.69666e11i		−	0.857220i
$668$		−	7.10883e10i		−	0.357020i
$669$		0			0
$670$	−3.47185e11			−1.72291
$671$	−4.03908e9			−0.0199247
$672$		0			0
$673$		−	1.01126e11i		−	0.492948i	−0.969149	−	0.246474i	$-0.920728\pi$
$673$		−	1.01126e11i		−	0.492948i	0.969149	−	0.246474i	$-0.0792719\pi$
$674$	−1.41233e11			−0.684381
$675$		0			0
$676$	−6.95350e10			−0.332979
$677$		−	1.62812e11i		−	0.775053i	−0.921858	−	0.387527i	$-0.873329\pi$
$677$		−	1.62812e11i		−	0.775053i	0.921858	−	0.387527i	$-0.126671\pi$
$678$		0			0
$679$		−	4.58019e10i		−	0.215479i
$680$		−	6.68353e9i		−	0.0312587i
$681$		0			0
$682$	−1.25079e10			−0.0578158
$683$			4.24838e11i			1.95227i	0.217157	+	0.976137i	$0.430322\pi$
$683$			4.24838e11i			1.95227i	−0.217157	+	0.976137i	$0.569678\pi$
$684$		0			0
$685$	−4.67573e11			−2.12367
$686$		−	4.43648e10i		−	0.200328i
$687$		0			0
$688$	−3.86233e10			−0.172383
$689$	1.05483e11			0.468066
$690$		0			0
$691$	−4.07739e11			−1.78842			−0.894210	−	0.447647i	$-0.852262\pi$
$691$	−4.07739e11			−1.78842			−0.894210	+	0.447647i	$0.852262\pi$
$692$		−	8.40718e10i		−	0.366628i
$693$		0			0
$694$			2.53463e11i			1.09264i
$695$	1.18642e11			0.508511
$696$		0			0
$697$		−	1.52406e10i		−	0.0645761i
$698$		−	2.80077e11i		−	1.17993i
$699$		0			0
$700$	1.99887e10			0.0832514
$701$	6.60489e10			0.273523			0.136761	−	0.990604i	$-0.456331\pi$
$701$	6.60489e10			0.273523			0.136761	+	0.990604i	$0.456331\pi$
$702$		0			0
$703$	−2.63757e11	−	8.44704e10i	−1.07990	−	0.345847i
$704$	8.78767e9			0.0357753
$705$		0			0
$706$			8.58289e10i			0.345474i
$707$	−7.59017e9			−0.0303790
$708$		0			0
$709$	−8.16942e10			−0.323300			−0.161650	−	0.986848i	$-0.551682\pi$
$709$	−8.16942e10			−0.323300			−0.161650	+	0.986848i	$0.551682\pi$
$710$	2.54135e11			1.00007
$711$		0			0
$712$	1.03608e11			0.403154
$713$		−	9.24123e10i		−	0.357579i
$714$		0			0
$715$			6.35865e10i			0.243299i
$716$		−	1.50493e11i		−	0.572617i
$717$		0			0
$718$		−	7.82349e10i		−	0.294376i
$719$	2.01728e11			0.754834			0.377417	−	0.926043i	$-0.376812\pi$
$719$	2.01728e11			0.754834			0.377417	+	0.926043i	$0.376812\pi$
$720$		0			0
$721$		−	4.90025e9i		−	0.0181333i
$722$	−1.11625e11	+	1.56398e11i	−0.410782	+	0.575550i
$723$		0			0
$724$		−	8.61530e10i		−	0.313557i
$725$			2.20133e11i			0.796770i
$726$		0			0
$727$	1.68993e11			0.604966			0.302483	−	0.953155i	$-0.402184\pi$
$727$	1.68993e11			0.604966			0.302483	+	0.953155i	$0.402184\pi$
$728$	−8.21446e9			−0.0292451
$729$		0			0
$730$		−	2.46165e11i		−	0.866830i
$731$	1.18351e10			0.0414480
$732$		0			0
$733$	3.42616e11			1.18684			0.593420	−	0.804893i	$-0.297777\pi$
$733$	3.42616e11			1.18684			0.593420	+	0.804893i	$0.297777\pi$
$734$		−	3.91955e11i		−	1.35037i
$735$		0			0
$736$			6.49261e10i			0.221263i
$737$			1.39879e11i			0.474115i
$738$		0			0
$739$	5.33253e11			1.78795			0.893975	−	0.448117i	$-0.147905\pi$
$739$	5.33253e11			1.78795			0.893975	+	0.448117i	$0.147905\pi$
$740$			2.50063e11i			0.833915i
$741$		0			0
$742$	−2.48430e10			−0.0819576
$743$		−	3.52979e10i		−	0.115823i	−0.998322	−	0.0579113i	$-0.981556\pi$
$743$		−	3.52979e10i		−	0.115823i	0.998322	−	0.0579113i	$-0.0184441\pi$
$744$		0			0
$745$	−7.12234e11			−2.31205
$746$	1.50128e11			0.484737
$747$		0			0
$748$	−2.69276e9			−0.00860185
$749$		−	1.80258e9i		−	0.00572754i
$750$		0			0
$751$			4.77175e11i			1.50009i	0.661386	+	0.750045i	$0.269968\pi$
$751$			4.77175e11i			1.50009i	−0.661386	+	0.750045i	$0.730032\pi$
$752$	−1.19255e10			−0.0372911
$753$		0			0
$754$		−	9.04650e10i		−	0.279895i
$755$		−	6.56073e11i		−	2.01913i
$756$		0			0
$757$	3.54055e11			1.07817			0.539084	−	0.842252i	$-0.318770\pi$
$757$	3.54055e11			1.07817			0.539084	+	0.842252i	$0.318770\pi$
$758$	1.56007e11			0.472572
$759$		0			0
$760$	1.65224e11	+	5.29145e10i	0.495245	+	0.158606i
$761$	3.22849e11			0.962632			0.481316	−	0.876547i	$-0.340159\pi$
$761$	3.22849e11			0.962632			0.481316	+	0.876547i	$0.340159\pi$
$762$		0			0
$763$			6.54574e10i			0.193135i
$764$	−3.19744e11			−0.938488
$765$		0			0
$766$	−2.51053e11			−0.729205
$767$	−2.54181e11			−0.734449
$768$		0			0
$769$	8.99891e10			0.257326			0.128663	−	0.991688i	$-0.458931\pi$
$769$	8.99891e10			0.257326			0.128663	+	0.991688i	$0.458931\pi$
$770$		−	1.49757e10i		−	0.0426013i
$771$		0			0
$772$		−	3.05126e11i		−	0.859033i
$773$			3.43312e11i			0.961548i	0.876845	+	0.480774i	$0.159644\pi$
$773$			3.43312e11i			0.961548i	−0.876845	+	0.480774i	$0.840356\pi$
$774$		0			0
$775$			1.19900e11i			0.332363i
$776$	−1.93023e11			−0.532306
$777$		0			0
$778$			5.95119e10i			0.162437i
$779$	3.76765e11	+	1.20662e11i	1.02311	+	0.327659i
$780$		0			0
$781$		−	1.02390e11i		−	0.275203i
$782$		−	1.98950e10i		−	0.0532006i
$783$		0			0
$784$	−9.25159e10			−0.244879
$785$	−7.02067e11			−1.84884
$786$		0			0
$787$			2.82012e11i			0.735137i	0.929997	+	0.367568i	$0.119810\pi$
$787$			2.82012e11i			0.735137i	−0.929997	+	0.367568i	$0.880190\pi$
$788$	−5.22729e10			−0.135573
$789$		0			0
$790$	−5.66851e11			−1.45533
$791$			7.03446e10i			0.179690i
$792$		0			0
$793$		−	1.59116e10i		−	0.0402365i
$794$			9.46857e10i			0.238233i
$795$		0			0
$796$	2.55390e11			0.636139
$797$			6.48442e11i			1.60708i	0.595250	+	0.803540i	$0.297053\pi$
$797$			6.48442e11i			1.60708i	−0.595250	+	0.803540i	$0.702947\pi$
$798$		0			0
$799$	3.65427e9			0.00896631
$800$		−	8.42381e10i		−	0.205659i
$801$		0			0
$802$	4.29904e11			1.03914
$803$	−9.91785e10			−0.238537
$804$		0			0
$805$	1.10645e11			0.263480
$806$		−	4.92736e10i		−	0.116755i
$807$		0			0
$808$			3.19872e10i			0.0750466i
$809$	−6.90473e11			−1.61195			−0.805977	−	0.591947i	$-0.798359\pi$
$809$	−6.90473e11			−1.61195			−0.805977	+	0.591947i	$0.798359\pi$
$810$		0			0
$811$		−	1.59384e11i		−	0.368434i	−0.982886	−	0.184217i	$-0.941025\pi$
$811$		−	1.59384e11i		−	0.368434i	0.982886	−	0.184217i	$-0.0589750\pi$
$812$			2.13060e10i			0.0490092i
$813$		0			0
$814$	1.00749e11			0.229479
$815$	−5.27561e11			−1.19576
$816$		0			0
$817$	−9.37005e10	+	2.92578e11i	−0.210307	+	0.656679i
$818$	−4.72977e10			−0.105640
$819$		0			0
$820$		−	3.57204e11i		−	0.790061i
$821$	7.47845e11			1.64603			0.823017	−	0.568016i	$-0.192289\pi$
$821$	7.47845e11			1.64603			0.823017	+	0.568016i	$0.192289\pi$
$822$		0			0
$823$	7.29538e10			0.159019			0.0795095	−	0.996834i	$-0.474665\pi$
$823$	7.29538e10			0.159019			0.0795095	+	0.996834i	$0.474665\pi$
$824$	−2.06511e10			−0.0447955
$825$		0			0
$826$	5.98637e10			0.128601
$827$		−	1.71778e11i		−	0.367236i	−0.982998	−	0.183618i	$-0.941219\pi$
$827$		−	1.71778e11i		−	0.367236i	0.982998	−	0.183618i	$-0.0587809\pi$
$828$		0			0
$829$		−	8.66395e11i		−	1.83442i	−0.398408	−	0.917208i	$-0.630437\pi$
$829$		−	8.66395e11i		−	1.83442i	0.398408	−	0.917208i	$-0.369563\pi$
$830$			4.28761e11i			0.903447i
$831$		0			0
$832$			3.46182e10i			0.0722455i
$833$	2.83492e10			0.0588790
$834$		0			0
$835$			5.10546e11i			1.05024i
$836$	2.13190e10	−	6.65681e10i	0.0436457	−	0.136283i
$837$		0			0
$838$			7.70698e10i			0.156282i
$839$			2.46507e11i			0.497487i	0.968569	+	0.248743i	$0.0800176\pi$
$839$			2.46507e11i			0.497487i	−0.968569	+	0.248743i	$0.919982\pi$
$840$		0			0
$841$	2.65606e11			0.530951
$842$	5.38065e10			0.107050
$843$		0			0
$844$			2.82989e11i			0.557698i
$845$	4.99391e11			0.979521
$846$		0			0
$847$	6.76264e10			0.131396
$848$			1.04696e11i			0.202463i
$849$		0			0
$850$			2.58127e10i			0.0494490i
$851$			7.44365e11i			1.41928i
$852$		0			0
$853$	−6.05188e11			−1.14313			−0.571563	−	0.820558i	$-0.693663\pi$
$853$	−6.05188e11			−1.14313			−0.571563	+	0.820558i	$0.693663\pi$
$854$			3.74743e9i			0.00704534i
$855$		0			0
$856$	−7.59661e9			−0.0141490
$857$		−	8.23654e11i		−	1.52694i	−0.645844	−	0.763469i	$-0.723495\pi$
$857$		−	8.23654e11i		−	1.52694i	0.645844	−	0.763469i	$-0.276505\pi$
$858$		0			0
$859$	−3.99209e11			−0.733209			−0.366605	−	0.930377i	$-0.619480\pi$
$859$	−3.99209e11			−0.733209			−0.366605	+	0.930377i	$0.619480\pi$
$860$	2.77387e11			0.507098
$861$		0			0
$862$	−6.21236e11			−1.12519
$863$		−	6.65968e11i		−	1.20063i	−0.799762	−	0.600317i	$-0.795041\pi$
$863$		−	6.65968e11i		−	1.20063i	0.799762	−	0.600317i	$-0.204959\pi$
$864$		0			0
$865$			6.03792e11i			1.07851i
$866$	6.41490e11			1.14056
$867$		0			0
$868$			1.16047e10i			0.0204436i
$869$			2.28381e11i			0.400480i
$870$		0			0
$871$	−5.51041e11			−0.957439
$872$	2.75857e11			0.477109
$873$		0			0
$874$	4.91826e11	+	1.57512e11i	0.842880	+	0.269940i
$875$	−2.01609e10			−0.0343936
$876$		0			0
$877$		−	1.42492e11i		−	0.240875i	−0.992721	−	0.120438i	$-0.961570\pi$
$877$		−	1.42492e11i		−	0.240875i	0.992721	−	0.120438i	$-0.0384297\pi$
$878$	−7.55174e11			−1.27078
$879$		0			0
$880$	−6.31118e10			−0.105240
$881$	6.50983e10			0.108060			0.0540302	−	0.998539i	$-0.482793\pi$
$881$	6.50983e10			0.108060			0.0540302	+	0.998539i	$0.482793\pi$
$882$		0			0
$883$	3.09709e11			0.509462			0.254731	−	0.967012i	$-0.418013\pi$
$883$	3.09709e11			0.509462			0.254731	+	0.967012i	$0.418013\pi$
$884$		−	1.06079e10i		−	0.0173708i
$885$		0			0
$886$		−	4.90864e10i		−	0.0796574i
$887$			4.50470e11i			0.727732i	0.931451	+	0.363866i	$0.118543\pi$
$887$			4.50470e11i			0.727732i	−0.931451	+	0.363866i	$0.881457\pi$
$888$		0			0
$889$		−	4.29112e9i		−	0.00687012i
$890$	−7.44095e11			−1.18595
$891$		0			0
$892$			1.29210e11i			0.204096i
$893$	−2.89314e10	+	9.03377e10i	−0.0454950	+	0.142057i
$894$		0			0
$895$			1.08082e12i			1.68446i
$896$		−	8.15314e9i		−	0.0126501i
$897$		0			0
$898$	−8.12122e11			−1.24887
$899$	−1.27802e11			−0.195658
$900$		0			0
$901$		−	3.20815e10i		−	0.0486805i
$902$	−1.43916e11			−0.217411
$903$		0			0
$904$	2.96453e11			0.443897
$905$			6.18739e11i			0.922387i
$906$		0			0
$907$		−	5.45609e11i		−	0.806218i	−0.915152	−	0.403109i	$-0.867929\pi$
$907$		−	5.45609e11i		−	0.806218i	0.915152	−	0.403109i	$-0.132071\pi$
$908$			1.37255e11i			0.201922i
$909$		0			0
$910$	5.89951e10			0.0860301
$911$			5.13209e11i			0.745111i	0.928010	+	0.372556i	$0.121518\pi$
$911$			5.13209e11i			0.745111i	−0.928010	+	0.372556i	$0.878482\pi$
$912$		0			0
$913$	1.72746e11			0.248613
$914$		−	9.12216e11i		−	1.30711i
$915$		0			0
$916$	2.79989e11			0.397703
$917$	5.68832e10			0.0804464
$918$		0			0
$919$	−1.31908e10			−0.0184930			−0.00924652	−	0.999957i	$-0.502943\pi$
$919$	−1.31908e10			−0.0184930			−0.00924652	+	0.999957i	$0.502943\pi$
$920$		−	4.66290e11i		−	0.650886i
$921$		0			0
$922$		−	6.25875e10i		−	0.0866092i
$923$	4.03355e11			0.555751
$924$		0			0
$925$		−	9.65774e11i		−	1.31919i
$926$		−	4.89264e11i		−	0.665426i
$927$		0			0
$928$	8.97896e10			0.121069
$929$	6.52670e11			0.876256			0.438128	−	0.898913i	$-0.355642\pi$
$929$	6.52670e11			0.876256			0.438128	+	0.898913i	$0.355642\pi$
$930$		0			0
$931$	−2.24445e11	+	7.00823e11i	−0.298752	+	0.932845i
$932$	−2.04882e11			−0.271544
$933$		0			0
$934$			2.82178e11i			0.370797i
$935$	1.93390e10			0.0253040
$936$		0			0
$937$	−4.27565e11			−0.554681			−0.277341	−	0.960772i	$-0.589453\pi$
$937$	−4.27565e11			−0.554681			−0.277341	+	0.960772i	$0.589453\pi$
$938$	1.29779e11			0.167646
$939$		0			0
$940$	8.56473e10			0.109699
$941$			1.36175e12i			1.73676i	0.495898	+	0.868381i	$0.334839\pi$
$941$			1.36175e12i			1.73676i	−0.495898	+	0.868381i	$0.665161\pi$
$942$		0			0
$943$		−	1.06329e12i		−	1.34464i
$944$		−	2.52283e11i		−	0.317688i
$945$		0			0
$946$		−	1.11758e11i		−	0.139545i
$947$	1.00481e12			1.24935			0.624675	−	0.780885i	$-0.285232\pi$
$947$	1.00481e12			1.24935			0.624675	+	0.780885i	$0.285232\pi$
$948$		0			0
$949$		−	3.90704e11i		−	0.481707i
$950$	−6.38118e11	−	2.04363e11i	−0.783441	−	0.250904i
$951$		0			0
$952$			2.49833e9i			0.00304160i
$953$			3.27699e11i			0.397286i	0.980072	+	0.198643i	$0.0636534\pi$
$953$			3.27699e11i			0.397286i	−0.980072	+	0.198643i	$0.936347\pi$
$954$		0			0
$955$	2.29636e12			2.76074
$956$	3.10399e11			0.371611
$957$		0			0
$958$			6.52738e11i			0.774956i
$959$	1.74780e11			0.206642
$960$		0			0
$961$	7.83281e11			0.918384
$962$			3.96891e11i			0.463416i
$963$		0			0
$964$		−	2.55455e11i		−	0.295805i
$965$			2.19137e12i			2.52701i
$966$		0			0
$967$	1.10076e12			1.25888			0.629442	−	0.777048i	$-0.283284\pi$
$967$	1.10076e12			1.25888			0.629442	+	0.777048i	$0.283284\pi$
$968$		−	2.84997e11i		−	0.324593i
$969$		0			0
$970$	1.38626e12			1.56588
$971$		−	8.23490e11i		−	0.926363i	−0.886263	−	0.463182i	$-0.846708\pi$
$971$		−	8.23490e11i		−	0.926363i	0.886263	−	0.463182i	$-0.153292\pi$
$972$		0			0
$973$	−4.43489e10			−0.0494802
$974$	4.45764e11			0.495301
$975$		0			0
$976$	1.57928e10			0.0174044
$977$		−	1.22250e12i		−	1.34174i	−0.741574	−	0.670871i	$-0.765920\pi$
$977$		−	1.22250e12i		−	1.34174i	0.741574	−	0.670871i	$-0.234080\pi$
$978$		0			0
$979$			2.99792e11i			0.326354i
$980$	6.64436e11			0.720359
$981$		0			0
$982$		−	1.22384e12i		−	1.31607i
$983$		−	1.79480e12i		−	1.92222i	−0.276167	−	0.961110i	$-0.589064\pi$
$983$		−	1.79480e12i		−	1.92222i	0.276167	−	0.961110i	$-0.410936\pi$
$984$		0			0
$985$	3.75417e11			0.398813
$986$	−2.75138e10			−0.0291100
$987$		0			0
$988$	2.62238e11	+	8.39841e10i	0.275213	+	0.0881393i
$989$	8.25702e11			0.863054
$990$		0			0
$991$		−	1.61330e11i		−	0.167271i	−0.996496	−	0.0836355i	$-0.973347\pi$
$991$		−	1.61330e11i		−	0.167271i	0.996496	−	0.0836355i	$-0.0266532\pi$
$992$	4.89058e10			0.0505026
$993$		0			0
$994$	−9.49966e10			−0.0973111
$995$	−1.83418e12			−1.87132
$996$		0			0
$997$	6.29339e11			0.636948			0.318474	−	0.947932i	$-0.396830\pi$
$997$	6.29339e11			0.636948			0.318474	+	0.947932i	$0.396830\pi$
$998$		−	1.01492e12i		−	1.02308i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	342.9.d.a.37.6		12
3.2	odd	2		38.9.b.a.37.10	yes	12
12.11	even	2		304.9.e.e.113.6		12
19.18	odd	2	inner	342.9.d.a.37.12		12
57.56	even	2		38.9.b.a.37.3	✓	12
228.227	odd	2		304.9.e.e.113.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.3	✓	12	57.56	even	2
38.9.b.a.37.10	yes	12	3.2	odd	2
304.9.e.e.113.6		12	12.11	even	2
304.9.e.e.113.7		12	228.227	odd	2
342.9.d.a.37.6		12	1.1	even	1	trivial
342.9.d.a.37.12		12	19.18	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	342.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -128.000 q^{4} +919.278 q^{5} -343.629 q^{7} +1448.15i q^{8} +O(q^{10})\)	\(q-11.3137i q^{2} -128.000 q^{4} +919.278 q^{5} -343.629 q^{7} +1448.15i q^{8} -10400.4i q^{10} -4190.29 q^{11} -16507.2i q^{13} +3887.72i q^{14} +16384.0 q^{16} -5020.47 q^{17} +(39747.8 - 124112. i) q^{19} -117668. q^{20} +47407.7i q^{22} -350263. q^{23} +454447. q^{25} -186758. q^{26} +43984.5 q^{28} +484397. i q^{29} +263837. i q^{31} -185364. i q^{32} +56800.1i q^{34} -315891. q^{35} -2.12516e6i q^{37} +(-1.40416e6 - 449695. i) q^{38} +1.33126e6i q^{40} +3.03570e6i q^{41} -2.35738e6 q^{43} +536357. q^{44} +3.96277e6i q^{46} -727875. q^{47} -5.64672e6 q^{49} -5.14148e6i q^{50} +2.11293e6i q^{52} +6.39013e6i q^{53} -3.85204e6 q^{55} -497628. i q^{56} +5.48032e6 q^{58} -1.53982e7i q^{59} +963914. q^{61} +2.98497e6 q^{62} -2.09715e6 q^{64} -1.51747e7i q^{65} -3.33818e7i q^{67} +642620. q^{68} +3.57390e6i q^{70} +2.44350e7i q^{71} +2.36687e7 q^{73} -2.40434e7 q^{74} +(-5.08772e6 + 1.58863e7i) q^{76} +1.43991e6 q^{77} -5.45025e7i q^{79} +1.50615e7 q^{80} +3.43450e7 q^{82} -4.12252e7 q^{83} -4.61521e6 q^{85} +2.66707e7i q^{86} -6.06818e6i q^{88} -7.15445e7i q^{89} +5.67237e6i q^{91} +4.48337e7 q^{92} +8.23496e6i q^{94} +(3.65393e7 - 1.14093e8i) q^{95} +1.33289e8i q^{97} +6.38853e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7}+O(q^{10}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7	\( 12 q - 1536 q^{4} - 558 q^{5} - 5422 q^{7} + 12546 q^{11} + 196608 q^{16} - 270810 q^{17} + 41512 q^{19} + 71424 q^{20} + 823956 q^{23} + 865538 q^{25} + 431616 q^{26} + 694016 q^{28} + 1194378 q^{35} - 998784 q^{38} + 7586646 q^{43} - 1605888 q^{44} + 20260530 q^{47} - 19498842 q^{49} - 14858554 q^{55} - 5506560 q^{58} - 41363266 q^{61} - 32266752 q^{62} - 25165824 q^{64} + 34663680 q^{68} + 87906498 q^{73} + 2149632 q^{74} - 5313536 q^{76} + 78817962 q^{77} - 9142272 q^{80} - 49609728 q^{82} + 55944960 q^{83} + 25440254 q^{85} - 105466368 q^{92} - 81396774 q^{95}+O(q^{100}) \) 12 * q - 1536 * q^4 - 558 * q^5 - 5422 * q^7 + 12546 * q^11 + 196608 * q^16 - 270810 * q^17 + 41512 * q^19 + 71424 * q^20 + 823956 * q^23 + 865538 * q^25 + 431616 * q^26 + 694016 * q^28 + 1194378 * q^35 - 998784 * q^38 + 7586646 * q^43 - 1605888 * q^44 + 20260530 * q^47 - 19498842 * q^49 - 14858554 * q^55 - 5506560 * q^58 - 41363266 * q^61 - 32266752 * q^62 - 25165824 * q^64 + 34663680 * q^68 + 87906498 * q^73 + 2149632 * q^74 - 5313536 * q^76 + 78817962 * q^77 - 9142272 * q^80 - 49609728 * q^82 + 55944960 * q^83 + 25440254 * q^85 - 105466368 * q^92 - 81396774 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more